论文标题
在算术功能的迭代中增加了降低模式
Increasing-decreasing patterns in the iteration of an arithmetic function
论文作者
论文摘要
令$ω$为一组正整数，让$ f：ω\rightarrowΩ$为算术函数。令$ v =（v_i）_ {i = 1}^n $是正整数的有限顺序。如果，对于$ f $，对于$ f $，则一个整数$ m \具有\ textIt {增加decreasing模式} $ v $。 v_ {i-1} +1}（m）<\ cdots <f^{v_1+ \ cdots+ v_ {i-1}+ v_ {i}}}}（m）\]，对于所有整数$ i \ in \ in \ in \ in \ in \ in \ {2，\ ldots v_ {i-1}}（m）> f^{v_1+\ cdots+v_ {i-1} +1} +1}（m）> \ cdots> f^{v_1+\ cdots+cdots+v_ {i-1}+v_i}（m）。 \ \]算术函数$ f $是\ textit {griang flical in decreasing}如果对于每个有限序列$ v $的正整数$ v $，则存在一个整数$ m \ inω$，以至于$ m $具有增加消除模式$ v $，而不是$ f $。本文证明了锡拉丘兹的功能正在大大提高折扣。
Let $Ω$ be a set of positive integers and let $f:Ω\rightarrow Ω$ be an arithmetic function. Let $V = (v_i)_{i=1}^n$ be a finite sequence of positive integers. An integer $m \in Ω$ has \textit{increasing-decreasing pattern} $V$ with respect to $f$ if, for all odd integers $i \in \{1,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) < f^{v_1+ \cdots + v_{i-1}+1}(m) < \cdots < f^{v_1+ \cdots + v_{i-1}+v_{i}}(m) \] and, for all even integers $i \in \{2,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) > f^{v_1+ \cdots +v_{i-1}+1}(m) > \cdots > f^{v_1+ \cdots +v_{i-1}+v_i}(m). \] The arithmetic function $f$ is \textit{wildly increasing-decreasing} if, for every finite sequence $V$ of positive integers, there exists an integer $m \in Ω$ such that $m$ has increasing-decreasing pattern $V$ with respect to $f$. This paper gives a proof that the Syracuse function is wildly increasing-decreasing.
